As outlined above, the components of the \(\lambda ^{[i]}_\tau (\mathbb {G}_m)\) -fixed locus are products of two Quot schemes, corresponding to the two sheaves in the associated graded of the filtration induced by this 1-PS, and indexed by pairs of Hilbert polynomials \((P^{\prime },P^{\prime \prime })\)  such that \(P = P^{\prime } + P^{\prime \prime }\) . One such component is given by the pair \((P_{\le i}, P_{>i})\)  and we claim this is the minimal weight space. Since \(Y_{\tau }^{ss}\)  is open in \(X\) , the minimal weight for \(\lambda _{\tau }^{[i]}\)  acting on \(X\)  is the same as the minimal weight for the action on \(Y_{\tau }^{ss}\) . For any \(q_\mathcal {E}\in Y_{\tau }^{ss}\) , a quick calculation shows the Hilbert–Mumford weight of \(\lambda ^{[i]}\)  with respect to the twisted linearisation \(\mathcal {L}_\tau \)  is equal to zero; this computation follows from [1]} and is similar to [2]}. By taking \(m\)  sufficiently large, we can ensure that the only pair of polynomials \((P^{\prime },P^{\prime \prime })\)  giving this minimum weight is \((P_{\le i}, P_{>i})\) .
The final statement follows, as any sheaf quotient sheaf in \(U_\tau ^{[i]}Z_{\tau ,\min }^{[i]} =P_\tau Z_{\tau ,\min }^{[i]}\)  is isomorphic to a sheaf in \(Z_{\min }^{[i]}\)  and this is precisely the locus where \(\mathcal {E}^{(i)} \subset \mathcal {E}\)  is split.
